The convergence of uniform random quadrangulations towards the Brownian map [1]}, [2]} gives a rough lower bound on the size of macroscopic bottlenecks in finite quadrangulations.
Let us be more precise. Fix \(\delta >0\) . Since the Brownian map is homeomorphic to the sphere [3]}, we can find \(\varepsilon >0\)  such that with probability close to 1, for \(n\)  large enough, any cycle in \(Q_n\)  that separates \(Q_n\)  in two subsets, each with at least \(\delta n\)  faces, must have length at least \(\varepsilon n^{1/4}\) . This is the best one can expect: with high probability it is possible to find sets of size roughly \(n/2\)  and perimeter no larger than some large constant times \(n^{1/4}\) .
However, this result only gives a lower bound on the size of bottlenecks at large scales (where the infimum holds over subsets \(S\)  of \(F(Q_n)\)  with \(\delta n \le |S| \le n/2\) ).
